2. Main Results

ON THE p-DOMINATION NUMBER OF CACTUS GRAPHS

Mostafa Blidia Mustapha Chellali

Department of Mathematics, University of Blida B.P. 270, Blida, Algeria

e-mail: mblidia@hotmail.com e-mail: mchellali@hotmail.com

and Lutz Volkmann

Lehrstuhl II f¨ur Mathematik, RWTH Aachen Templergraben 55, D–52056 Aachen, Germany

e-mail: volkm@math2.rwth-aachen.de

Abstract

Let p be a positive integer and G = (V, E) a graph. A subset S of V is a p-dominating set if every vertex of V − S is dominated at least p times. The minimum cardinality of a p-dominating set a of G is the p-domination number γp(G). It is proved for a cactus graph G that γp(G) 6 (|V | + |Lp(G)| + c(G))/2, for every positive integer p > 2, where Lp(G) is the set of vertices of G of degree at most p − 1 and c(G) is the number of odd cycles in G.

Keywords: p-domination number, cactus graphs.

2000 Mathematics Subject Classification: 05C69.

1. Introduction

Let G = (V (G), E(G)) be a graph with vertex set V (G) and edge set E(G).

The order of G is n(G) = |V (G)| and the degree of a vertex v, denoted by deg_G(v), is the number of vertices adjacent to v. A vertex of degree one is called a leaf and its neighbor is called a support vertex. A vertex of V is called a cut vertex if removing it from G increases the number of components of G. A graph G is called a cactus graph if each edge of G is contained in at most one cycle. A unicycle graph is a graph with exactly one cycle. A tree T is a double star if it contains exactly two vertices that are not leaves.

A double star with, respectively p and q leaves attached at each support vertex, is denoted by S_p,q.

For a positive integer p, a subset S of V (G) is a p-dominating set if every vertex not in S is adjacent to at least p vertices of S. The p-domination number γ_p(G) is the minimum cardinality of a p-dominating set of G. Note that every graph G has a p-dominating set, since V (G) is such a set. Also the 1-domination number γ₁(G) is the usual domination number γ(G). The concept of p-domination was introduced by Fink and Jacobson [2, 3]. For more details on domination and its variations see the books of Haynes, Hedetniemi, and Slater [4, 5].

We make a straightforward observation.

Observation 1. Every p-dominating set of a graph G contains any vertex of degree at most p − 1.

In this paper we present an upper bound for the p-domination number for cactus graphs in terms of the order, the number of odd cycles and the number of vertices of degrees at most p − 1.

The following result due to Blidia et al. [1] will be useful for the next.

Let L_p(G) denote the set {x ∈ V (G) : deg_G(x) 6 p − 1}.

Theorem 2 (Blidia, Chellali and Volkmann [1]). Let p be a positive integer.

If G is a bipartite graph then

γ_p(G) 6 (n + |L_p(G)|)/2 .

2. Main Results

We begin by giving an upper bound for the p-domination number for con- nected unicycle graphs.

Theorem 3. Let p > 2 be a positive integer. If G is a connected unicycle graph then

γ_p(G) 6 (n + |L_p(G)| + 1)/2 and this bound is sharp.

P roof. Let G be a connected unicycle graph. If G is bipartite then the result is valid by Theorem 2. So assume that G contains an odd cycle denoted by C. If G = C, then γ_p(G) = n if p > 3 and γ_p(G) = (n + 1)/2 if p = 2, in both cases the result holds. Thus we assume that G 6= C, that is G contains at least one leaf.

Suppose that the result does not hold and let G be the smallest con- nected unicycle graph such that γ_p(G) > (n + |L_p(G)| + 1)/2. We claim that every vertex on C has degree exactly p. Suppose to the contrary that there is a vertex x ∈ C such that deg_G(x) 6= p and let y be one of its two neighbors on C. Let G⁰ be the spanning graph of G obtained by removing the edge xy.

Then G⁰ is tree and so a bipartite graph. We also have |L_p(G⁰)| 6 |L_p(G)|+1 and n(G) = n(G⁰). According to Theorem 2, we deduce that

γ_p(G) 6 γ_p(G⁰) 6 (n(G⁰) +¯

¯L_p(G⁰)¯

¯)/2 6 (n(G) + |L_p(G)| + 1)/2, a contradiction with our assumption.

Similarly, we will show that every vertex not on C and different to a leaf has degree at least p. Assume to the contrary that there is a vertex x ∈ V (G) − C different to a leaf with deg_G(x) 6 p − 1 and let z be its neighbor in the unique path from x to C. Let G₁ be the connected unicycle subgraph of G containing x and obtained by removing all the edges incident to x excepted the edge xz, and let G₂ be the component containing x by removing the edge xz. Let D₁ and D₂ denote a γ_p(G₁)-set and a γ_p(G₂)- set, respectively. Clearly G₁ contains C and G₂ is a tree, x ∈ D₁ ∩ D₂, x ∈ L_p(G₁)∩L_p(G₂), |L_p(G₁)|+|L_p(G₂)| = |L_p(G)|+1 and n(G₁)+n(G₂) = n(G) + 1. Furthermore, D₁∪ D₂ is a p-dominating set of G. In addition, G₁ and G₂ have order less than G and so satisfy the theorem, implying that

γ_p(G) 6 |D₁∪ D₂| = γ_p(G₁) + γ_p(G₂) − 1

6 (n(G₁) + |L_p(G₁)| + 1)/2 + (n(G₂) + |L_p(G₂)|)/2 − 1 6 (n + |L_p(G)| + 1)/2,

contradicting the assumption.

Suppose now that V (G) − C contains a support vertex. Let a be a support vertex of G of maximum distance from C. As seen above, a has degree at least p. Let G⁰ = G − (L_a∪ {a}). Then γ_p(G⁰) + |L_a| = γ_p(G), n(G⁰) = n(G) − |L_a| − 1 and L_p(G) > L_p(G⁰) + |L_a| − 1. It follows that

γ_p(G⁰) + |L_a| = γ_p(G) > (n(G) + |L_p(G)| + 1)/2 implying that

γ_p(G⁰) > (n(G) + |L_p(G)| + 1 − 2 |L_a|)/2 and so

γ_p(G⁰) > (n(G⁰) +¯

¯L_p(G⁰)¯

¯ + 1)/2

contradicting our assumption that G is the smallest graph that does not satisfy the theorem.

Consequently, every vertex of V (G) − C must be a leaf and so every vertex on C is adjacent to exactly p − 2 leaves, which implies that

γ_p(G) = n − (|V (C)| − 1)/2 = (n(G) + |L_p(G)| + 1)/2 a contradiction.

To see that this bound is sharp, consider the graph G formed by an odd cycle C where each vertex on C is adjacent to exactly p − 2 vertices. Then γ_p(G) = n − (|V (C)| − 1)/2 = (n(G) + |L_p(G)| + 1)/2.

Theorem 4. Let p > 2 be a positive integer. If G is a connected cactus graph with c(G) odd cycles then,

γ_p(G) 6 (n + |L_p(G)| + c(G))/2, and this bound is sharp.

P roof. If G is a bipartite graph, then by Theorem 2 the result holds. If G is a unicycle graph then by Theorem 3 the result is also valid. So consider a cactus graph G containing at least two cycles with one of odd length.

Assume that the result does not hold and let G be the smallest cactus graph such that γ_p(G) > (n(G) + |L_p(G)| + c(G))/2. We also assume that among all such graphs, G is the one having the fewest edges.

First, let u be a vertex on an odd cycle C of G and assume that deg_G(u) 6= p. Let G⁰ be the spanning graph of G obtained by removing an

edge of C incident with u. Then |L_p(G⁰)| 6 |L_p(G)|+1 and c(G⁰) = c(G)−1.

Also G⁰ satisfies the result and so γ_p(G) 6 γ_p(G⁰) 6 (n(G⁰) +¯

¯L_p(G⁰)¯

¯ + c(G⁰))/2

6 (n(G) + |L_p(G)| + 1 + c(G) − 1)/2 = (n + |L_p(G)| + c(G))/2 , a contradiction. Thus every vertex in an odd cycle has degree exactly p.

Now consider a vertex v different from a leaf and contained in no odd cycle. Then, either v is a cut vertex or v is on an even cycle and deg_G(v) = 2.

Suppose first that v is a cut vertex with deg_G(v) < p. Let G₁ and G₂ be two connected cactus subgraphs of G with V (G) = V (G₁) ∪ V (G₂) having v as a unique common vertex. Then, c(G) = c(G₁) + c(G₂), n(G) = n(G₁) + n(G₂) − 1, |L_p(G)| = |L_p(G₁)| + |L_p(G₂)| − 1. Now let D₁ and D₂ denote a γ_p(G₁)-set and a γ_p(G₂)-set, respectively. Then v ∈ D₁∪ D₂ and D₁ ∪ D₂ is a p-dominating set of G. Since G₁ and G₂ satisfy the result,

γ_p(G) 6 |D₁∪ D₂| = |D₁| + |D₂| − 1

6 (n(G₁) + |L_p(G₁)| + c(G₁))/2 + (n(G₂) + |L_p(G₂)| + c(G₂))/2 − 1 6 (n(G) + |L_p(G)| + c(G))/2 ,

a contradiction. Consequently, every cut vertex contained in no odd cycle has degree at least p.

Now let v be a vertex on an even cycle with deg_G(v) = 2. Since we have assumed in the beginning of the proof that G has at least two cycles, we have p > 3. We claim that each neighbor of v has degree exactly p. Indeed, let u be a neighbor of v and assume that deg_G(u) 6= p. Then every γ_p(G⁰)-set S is a p-dominating set of G where G⁰ is obtained from G by removing the edge vu. So

γ_p(G) 6 |S| 6 (n(G⁰) +¯

¯L_p(G⁰)¯

¯ + c(G⁰))/2 = (n(G) + |L_p(G)| + c(G))/2 , a contradiction. Thus deg_G(u) = p.

Now let C denote an odd cycle of length at least 5 and let w be a vertex on C, a and b its neighbors on C. Delete the edges wa, wb. The remaining graph has two components for otherwise wa or wb would be contained in two cycles. Let G₁be the component containing w and G₂ the other component where a new edge is added joining a and b. Then both G₁and G₂ verify the theorem. Also deg_G2(a) = deg_G2(b) = p, |L_p(G₁)| + |L_p(G₂)| 6 |L_p(G)| + 1

and c(G₁) + c(G₂) = c(G) − 1. Let D₁ and D₂ be a γ_p(G₁)-set and a γ_p(G₂)- set, respectively. Then D₁ contains w since deg_G1(w) = p − 2. It can be checked that D₁∪ D₂ is a p-dominating set of G. It follows that

γ_p(G) 6 |D₁∪ D₂|

6 (n(G₁) + |L_p(G₁)| + c(G₁))/2 + (n(G₂) + |L_p(G₂)| + c(G₂))/2 6 (n(G) + |L_p(G)| + 1 + c(G) − 1)/2 = (n(G) + |L_p(G)| + c(G))/2 contradicting our assumption. Thus it remains to investigate the case that each odd cycle is a triangle.

Let C = uvw be a triangle of G. If p = 2 then as claimed before G = C₃ and the theorem is valid. So assume that p > 3. Let G_u, G_v and G_w be the three components of G containing u, v, w, respectively, by removing the edges uv, uw and vw. Suppose that each component contains at most one vertex of degree at least p and let j the number of vertices of degree at least p in the three components. Then j 6 3 and |L_p(G)| = n − 3 − j. In this case, G_u is either a star of center vertex u with p − 2 leaves, or star of order at least 4 where u is a leaf if p = 3, or a double star S_p−3,p−1 with u as a support vertex if p > 4, or a graph formed by a cycle C₄ where u ∈ V (C₄) and is adjacent to p − 4 leaves (if p > 4), its neighbors on the cycle have degree 2 and the remaining vertex of the cycle is adjacent to p − 2 leaves.

Likewise G_v and G_w. If each component is a tree then G is a unicycle and the result follows by Theorem 3. So we assume that G_u is a component containing the cycle C₄. Now it is a routine matter to check that

γ_p(G) = n − (j + 1) 6 (n(G) + |L_p(G)| + c(G))/2 = n − 1 − j/2 , a contradiction.

Thus we may assume, without loss of generality, that G_u contains at least two vertices of degree at least p. Let G⁰ be the component containing v, w by removing the edges uv, uw. Let G₀ be the graph constructed from G⁰ by attaching v and w to the support vertices say a, b of a double star S_p−2,p−2 (so v, w, a, b induce a cycle C₄) and let D_u and D₀ a γ_p(G_u)-set and a γ_p(G₀)-set, respectively. Then, without loss of generality, D₀contains v, w, a all the leaves adjacent to a and b. Also D_u contains u since it has degree at most p−2. Obviously D_u∪(D₀−({a}∪L_a∪L_b)) is a p-dominating set of G. It is easy to check that G_u contains at least 2p − 1 vertices. Thus G₀ has order less than G since we have added 2p − 2 vertices and so both

G_u, G₀ verify the result. On the other hand, n(G) = n(G_u) + n(G₀) − 2p + 2, L_p(G) = L_p(G_u)−1+L_p(G₀)−2p+4, c(G) = c(G_u)+c(G₀)+1. Consequently

γ_p(G) 6 |D_u∪ (D₀− ({a} ∪ L_a∪ L_b))| = γ_p(G_u) + γ_p(G₀) − 2p + 3 6 (n(G_u) + |L_p(G_u)| + c(G_u))/2

+ (n(G₀) + |L_p(G₀)| + c(G₀))/2 − 2p + 3 6 (n(G) + |L_p(G)| + c(G))/2 ,

a contradiction with our assumption.

That this bound is sharp may be seen by considering the graph G_k formed by k > 1 triangles where each vertex of the triangle is attached to p − 2 leaves, and identifying a vertex of every triangle with a vertex of a path P_k. Then n(G_k) = (3p − 3)k, |L_p(G_k)| = 3(p − 2)k, c(G_k) = k and γ_p(G) = (n(G_k) + |L_p(G_k)| + c(G_k))/2 = (3p − 4)k.

References

[1] M. Blidia, M. Chellali and L. Volkmann, Some bounds on the p-domination number in trees, submitted for publication.

[2] J.F. Fink and M.S. Jacobson, n-domination in graphs, in: Y. Alavi and A.J.

Schwenk, eds, Graph Theory with Applications to Algorithms and Computer Science (Wiley, New York, 1985) 283–300.

[3] J.F. Fink and M.S. Jacobson, On n-domination, n-dependence and forbidden subgraphs, in: Graph Theory with Applications to Algorithms and Computer Science (Wiley, New York, 1985) 301–312.

[4] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1998).

[5] T.W. Haynes, S.T. Hedetniemi and P.J. Slater (eds), Domination in Graphs:

Advanced Topics (Marcel Dekker, New York, 1998).

Received 24 March 2004 Revised 26 August 2004